We would like to stress that when \(U\)  is exactly zero, one can argue that \(\left|\varphi _{1}\right\rangle \)  and \(\left|\varphi _{2}\right\rangle \)  are exponentially degenerate. The degeneracy originates
from the edge degree of freedom. Thus, any linear combination of them are
still edge states, with \(\left|\varphi _{1}\right\rangle \)  and \(\left|\varphi _{2}\right\rangle \)  being only one possible choice. Then
it does not make sense to discuss the behaviour of \(\left|\varphi _{1}\right\rangle \)  in real space when crossing the Dirac point, e.g. from
bottom-left to top-right, as a linear combination would mix left edge and
right edge. The naturalness of choosing \(\left|\varphi _{1}\right\rangle \)  and \(\left|\varphi _{2}\right\rangle \)  is seen as
following: when a small positive bias U, e.g. \(0.001eV\) , is included as
perturbation \(H_{\mathrm {per}}\) , the cross term \(\left\langle \varphi _{1}\right|H_{\mathrm {per}}\left|\varphi _{2}\right\rangle =0\) .
According to the first order degenerate perturbation theory [1]}, \(\left|\varphi _{1}\right\rangle \)  and \(\left|\varphi _{2}\right\rangle \)  are
the good states to use in perturbation. Energy of state in Eq. (REF ) would increase relatively to the energy of state in Eq. (REF ), as
the former has minor component on \(l=2\)  layer. Similarly, energy of state in
Eq. (REF ) would increase relatively to the energy of state in Eq. (REF ), as the latter has minor component on \(l=1\)  layer. Thus,\(\left|\varphi _{1}\right\rangle \) /\(\left|\varphi _{2}\right\rangle \)  would continuously evolve into Eq. (REF )/Eq. (REF ). Fig. REF  is plotted with \(U=0.001eV\)  and \(\gamma _{3}=0\) . The behaviour of
states is numerically smooth, indicating the degeneracy is effectively
lifted with a small bias.
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